Bounding Quantum-Classical Separations for Classes of Nonlocal Games

Stabilizer rank and higher-order Fourier analysis

“…We believe that the application we found motivate the further study of tools in higher-order Fourier analysis in quantum information theory. A work that connected to this area is [1], where the "Gowers Inverse Theorem" is utilised to analyze certain Bell inequalities. This current work is yet another example.…”

“…, x n ] as a map ι(P ) : F n p → T. Nonclassical polynomials that arise in this way are called classical polynomials and they are a subset of the nonclassical polynomials. Note that they take values in 1 p Z/Z. This confusing terminology has unfortunately become standard in the literature.…”

“…where • is entry-wise product of vectors. In the first equality we used the property in (1). Taking one more derivative…”

“…Indeed, P (x) = |x|/4 is a nonclassical polynomial of degree two. Note that it is not a classical polynomial since it takes values in 1 4 Z/Z.…”

“…Proposition 3.7 implies that its depth is at most 1. In other words, Q takes values in 1 4 Z/Z and so does Q. Similarly, for p > 2 a quadratic polynomial Q : H → F p on an affine subspace H ⊂ F n p , is simply a quadratic polynomial in local coordinates, i.e.…”